Consider a smooth function f : R n × R 1 × R 1 → R n, and let f x : R n × R 1 × R 1 → R n × n be the derivative with respect to the first argument. For fixed &ggr;, a turning point of f ( x , &lgr; &ggr; ) is an ( x , &lgr; ) such that f ( x , &lgr; , &ggr; ) = 0 and f ( x , &lgr; &ggr; ) has rank ( n - 1 ). The problem of computing such turning points has been extensively studied by the author, colleagues, and others in recent years. Under appropriate circumstances, there is a curve of such turning points parametrized by &ggr; containing a distinguished point at which 0 is an eigenvalue of f ( x , &lgr; &ggr; ) of algebraic multiplicity 2 and geometric multiplicity 1 . The task is to compute such a distinguished point: a BT-point.
The author summarizes the requisite theoretical background from other sources (some unfortunately nonarchival) to delineate assumptions that are necessary and convenient to characterize the object sought. An algorithm is formulated, a local convergence and rate of convergence theorem is proven, and two examples are discussed.
The paper is generally well written, though familiarity with the cited references is essential to grasp the details. The presentation is marred by an excessive number of linguistic and typographical errors, for which the editor and referees must accept partial responsibility.
